What is the inverse of the function $f(x)=\dfrac{-2x+2}{x+7}$ ? $ f^{-1}(x) =$
Answer: Let's start by replacing $f(x)$ with $y$. $y=\dfrac{-2x+2}{x+7}$ Now let's swap $x$ and $y$ and solve for $y$. $\dfrac{-2y+2}{y+7}=x$ [Why do we swap x and y?] $\begin{aligned} \dfrac{-2y+2}{y+7}&=x \\\\ -2y+2&=x(y+7) \\\\ -2y+2&=xy+7x \\\\ -2y-xy&=7x-2 \\\\ y(-2-x)&=7x-2 \\\\ y&=\dfrac{7x-2}{-2-x} \end{aligned}$ In conclusion, this is the inverse function: $f^{-1}(x)=\dfrac{7x-2}{-2-x}$ [I saw someone solve this problem by originally solving for x. Were they wrong?]